Finite Group Databasebeta

← #1625×C251 #44546212

Subgroups — #1625×C251 #44546212

Order 32128.

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#547×C251C211yesmaximal, characteristic, complemented
2#486×C251C211yesmaximal, characteristic, complemented
3#459×C251C211yesmaximal, characteristic, complemented
4#459×C251C211yesmaximal, characteristic, complemented
5#332×C251C211yesmaximal, characteristic, complemented
6D4⋊C8×C251C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#41×C251—21no—
9C2×D8×C251—21no—
10C2×D8×C251—21no—
11C4⋊Q8×C251C2211yescharacteristic
12#127×C251C2211yescharacteristic
13#123×C251—22no—
14C22⋊Q8×C251—21no—
15#121×C251—21no—
16C4×D4×C251C2211yescharacteristic
17C4×D4×C251C2211yescharacteristic
18C2×C4⋊C4×C251—21no—
19C4⋊C8×C251C2211yescharacteristic
20C4⋊C8×C251C2211yescharacteristic
21#102×C251—21no—
22#102×C251—21no—
23#102×C251—22no—
24C4×C8×C251C2211yescharacteristic
25#41×C251—42no—
26#41×C251—42no—
27C2×Q8×C251—21no—
28C2×D4×C251—22no—
29C2×D4×C251—21no—
30C2×D4×C251—21no—
31C2×D4×C251—21no—
32C2×D4×C251D411yescharacteristic
33C2×D4×C251D411yescharacteristic
34C22×C4×C251—22no—
35C22×C4×C251—21no—
36C22×C4×C251—21no—
37D8×C251—42no—
38D8×C251—42no—
39C2×C8×C251—21no—
40C2×C8×C251—21no—
41C2×C8×C251—21no—
42C4⋊C4×C251—22no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251D411yescharacteristic
45C4⋊C4×C251D411yescharacteristic
46C4⋊C4×C251D412yes—
47C22⋊C4×C251—22no—
48C22⋊C4×C251—21no—
49C22⋊C4×C251—21no—
50C42×C251C2311yescharacteristic
51C23×C251—21no—
52C23×C251—21no—
53C23×C251—41no—
54Q8×C251—41no—
55Q8×C251—41no—
56D4×C251—21no—
57D4×C251—21no—
58D4×C251—21no—
59D4×C251—42no—
60D4×C251—44no—
61D4×C251—41no—
62D4×C251—41no—
63D4×C251—41no—
64D4×C251—41no—
65D4×C251D812yescomplemented
66C2×C4×C251—21no—
67C2×C4×C251—21no—
68C2×C4×C251—22no—
69C2×C4×C251—44no—
70C2×C4×C251—41no—
71C2×C4×C251—41no—
72C2×C4×C251—41no—
73C2×C4×C251—41no—
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C2×C4×C251C2×D411yescharacteristic
77C8×C251—41no—
78C8×C251—41no—
79C8×C251—41no—
80C22×C251—22no—
81C22×C251—41no—
82C22×C251—41no—
83C22×C251—41no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251—81no—
87C22×C251—81no—
88C22×C251—81no—
89C22×C251#12011yescharacteristic
90C4×C251—21no—
91C4×C251—41no—
92C4×C251—42no—
93C4×C251—41no—
94C4×C251C2×D811yescharacteristic
95C4×C251#13511yescharacteristic
96C4×C251AGL(1,ℤ/8)11yescharacteristic
97C4×C251AGL(1,ℤ/8)11yescharacteristic
98C2×C251—42no—
99C2×C251—81no—
100C2×C251—161no—
101C2×C251#44711yescharacteristic
102C2×C251#44911yescharacteristic
103C2×C251#45411yescharacteristic
104C251#162511yescharacteristic, direct product factor, Sylow
105#1625C25111yesmaximal, characteristic, direct product factor, Sylow
106#547C2×C25111yescharacteristic, complemented
107#486C2×C25111yescharacteristic, complemented
108#459C2×C25111yescharacteristic, complemented
109#459C2×C25111yescharacteristic, complemented
110#332C2×C25111yescharacteristic, complemented
111D4⋊C8C2×C25111yescharacteristic, complemented
112D4⋊C8C2×C25111yescharacteristic, complemented
113C2×#41—21no—
114C2×D8—21no—
115C2×D8—21no—
116C4⋊Q8C22×C25111yescharacteristic
117#127C22×C25111yescharacteristic
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C22×C25111yescharacteristic
122C4×D4C22×C25111yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8C22×C25111yescharacteristic
125C4⋊C8C22×C25111yescharacteristic
126#102—21no—
127#102—21no—
128#102—22no—
129C4×C8C22×C25111yescharacteristic
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4D4×C25111yescharacteristic
138C2×D4D4×C25111yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—42no—
143D8—42no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4D4×C25111yescharacteristic
150C4⋊C4D4×C25111yescharacteristic
151C4⋊C4D4×C25112yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C23×C25111yescharacteristic
156C23—21no—
157C23—21no—
158C23—41no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—42no—
165D4—44no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4D8×C25112yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×D4×C25111yescharacteristic
180C2×C4C2×D4×C25111yescharacteristic
181C2×C4C2×D4×C25111yescharacteristic
182C8—41no—
183C8—41no—
184C8—41no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22#120×C25111yescharacteristic
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4C2×D8×C25111yescharacteristic
200C4#135×C25111yescharacteristic
201C4AGL(1,ℤ/8)×C25111yescharacteristic
202C4AGL(1,ℤ/8)×C25111yescharacteristic
203C2—42no—
204C2—81no—
205C2—161no—
206C2#447×C25111yescharacteristic
207C2#449×C25111yescharacteristic
208C2#454×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C42×C251 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #547×C251C2semidirect
2. #486×C251C2semidirect
3. #459×C251C2semidirect
4. #459×C251C2semidirect
5. #332×C251C2semidirect
6. D4⋊C8×C251C2semidirect
7. D4⋊C8×C251C2semidirect
65. D4×C251D8semidirect
104. C251#1625direct (N × H)
105. #1625C251direct (N × H)
106. #547C2×C251semidirect
107. #486C2×C251semidirect
108. #459C2×C251semidirect
109. #459C2×C251semidirect
110. #332C2×C251semidirect
111. D4⋊C8C2×C251semidirect
112. D4⋊C8C2×C251semidirect
170. D4D8×C251semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#547×C2512
#486×C2512
#459×C2512
#332×C2512
D4⋊C8×C2512
#1625251